A weighted residual relationship for the contact problem with Coulomb friction

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with Coulomb friction

Anh Le Van, T Nguyen

To cite this version:

Anh Le Van, T Nguyen. A weighted residual relationship for the contact problem with Coulomb

friction. Computers and Structures, Elsevier, 2009. �hal-03238939�

A. Le van1_{and T.T.H. Nguyen}

GeM (Laboratory of Civil and Mechanical Engineering), Faculty of Science - University of Nantes, 2, rue de la Houssiniere - BP 92208, 44322 Nantes Cedex 3, France

ABSTRACT

In this work, a weighted residual relationship is proposed as an extension of the standard virtual work principle to deal with the large deformation contact problem with Coulomb friction. This weak form is a mixed relationship involving the displacements and the multipliers defined on the reference contact surface of the contactor and is shown to be equivalent to the strong form of the initial/boundary value contact problem. The discretization in space by means of the finite element method is carried out on the mixed relationship in a simple way in order to obtain the semi-discrete equation system. The contact tangent stiffness is derived and numerical examples are presented to assess the efficiency of the formulation. Keywords : Contact; Friction; Weighted residual relationship; Finite element method.

1 Introduction

Problems where contact takes place between two or more bodies are widespread and of practical significance in me-chanics. Their solution is complex because of the nonlinear and non-smooth laws governing the interface response be-tween the contacting bodies. In the context of large deforma-tions and large slips, the problem becomes even more non-linear and robust formulations must be designed with a view to efficiently solve it.

Over more than three decades, a great number of formu-lations and solution methods have been proposed in the liter-ature in order to deal with different kinds of contact problems encountered, from frictionless contact in linear elastostatics up to frictional contact in large deformation with complex material constitutive and interfacial contact laws, either in statics or dynamics. The interested reader is referred to the book of Kikuchi and Oden [1] and the more recent books of Wriggers [2] and Laursen [3] for a review of the numerous references related to the existing methods for the treatment of the contact constraints, in particular the classical Lagrange multiplier, the penalty and the augmented Lagrangian meth-ods. As the formulation proposed in the present work is in-spired from methods of the augmented Lagrangian type, we shall limit ourselves to a brief review of these methods in the literature.

The augmented Lagrangian method is often chosen to cope with the contact inequality constraints inasmuch as it combines the regularizing effect of the penalty method and the exact satisfaction of the constraints ensured by the Lagrange multiplier method, without having the ill-conditioning problem inherent to the penalty method. It was applied to different contact problems in many variants. Kinematically linear problems with frictional contact were solved using the augmented Lagrangian approach by Alart and Curnier [4], Zavarise, Wriggers and Schrefler [5]. Large deformation contact without friction was also successfully addressed by the same method in the early work of Glowin-ski and Le Tallec [6] considering a plane rigid obstacle, in

Heegard and Curnier’s paper [7] considering contact between two elastic bodies, and later in Zavarise and Wriggers [8]. The solution of large deformation frictional contact problems between deformable bodies was accomplished by Simo and Laursen [9, 10] including the algorithmic symmetrization of the contact tangent stiffness matrix, Wriggers [11], Pietrzak and Curnier [12, 13] and more recently by Feng, Peyraut and Labed [14] for Blatz-Ko hyperelastic bodies, and Mi-jar and Arora [15, 16]. All previous works considered the Coulomb friction model or an extension of it to incorporate micro-mechanical effects for instance.

In most of the formulations, the tangential contact stresses were computed by locally integrating the contact laws and the contact stiffness matrix is consistently derived from these stresses, following the spirit of the return mapping scheme in elastoplasticity as described by Giannakopoulos in [17]. In some formulations, e.g. [5], the tangential contact law adopted was not of the rate type and the above-mentioned local integration was not necessary. Curnier and co-workers [4, 7, 13] chose another approach by directly incorporating the contact laws into their augmented Lagrangians, which enabled them to obtain the correct tangential contact stresses without resorting to the local integration.

As regards the global solution of the discrete equation system, Uzawa’s method was usually chosen because of its simplicity: the multipliers are updated in an outer augmenta-tion loop and the displacements computed in a standard way inside an inner loop while keeping the multipliers fixed. Oth-erwise, the discrete equation system can be solved simultane-ously for the displacements and the multipliers, as was done in [13, 8], for instance. One can also consult several different global algorithms for solving augmented Lagrangian prob-lems in [6], Chapters 3 and 7.

Motivation

The purpose of this paper is to propose a simple formu-lation for the contact problem with Coulomb friction which combines the benefits of the aforementioned formulations of the augmented Lagrangian type. The sought formulation is based on a new weak form – similar to the virtual work

ciple – which should fulfill the following conditions: (i) The weak form should incorporate the contact laws. More precisely, although it is an equality, it must imply the

inequality constraints specific to contact laws. The

conse-quence of this is that the contact laws are automatically satis-fied regardless of the constitutive laws of the contacting bod-ies and no local integration is needed at the contact surface level.

(ii) The weak form may look like an augmented La-grangian principle in that it involves both the displacements and the multipliers, but, more importantly, it should be stated in the form of a weighted residual relationship, with arbitrary (regular) virtual displacements and multipliers (an overview of the weighted residual methods can be found in [18]). As such, the weak form can be discretized by means of the finite element method in a straightforward way.

Outline of the paper

The paper is organized as follows. The strong form of

the initial/boundary value contact problem is summarized in Section 2. In Section 3, a weighted residual relationship is proposed as an extension of the standard virtual work prin-ciple with the aim of encompassing the large deformation contact response with Coulomb friction. This mixed weak form which involves both the displacements and the multipli-ers defined on the reference contact surface of the contactor will be shown to be equivalent to the strong form of the con-tact problem. Section 4 deals with the semi-discretization in space of the weak form by means of the finite element method, which results in a coupled nonlinear semi-discrete equation system in two unknowns: the nodal displacements and the nodal multipliers. In Section 5, the contact tangent stiffness matrix which is necessary to the Newton-Raphson iterative solution is derived by the exact linearization of the weak form. Different numerical examples assessing the effi-ciency of the proposed formulation are presented in Section 6.

2 Strong form of the contact problem

Consider two bodies whose positions at any time are represented by some regions in a three-dimensional Euclidean affine space. The position at any time of any particle belonging to the bodies is represented by a point in this space. As usually done, a fixed origin being chosen, any point as well as the associated position vector will be identified to a three-component vector in R3_.

We shall describe the mechanical problem using a Lagrangian description. The reference configuration of the two bodies are represented by the regions Ω(1)o and Ω(2)o in R3. Let us assume that in the time interval [O, T ] of interest, the two bodies

may come into contact with each other through the motions denoted φ(1)and φ(2)

∀i ∈ {1, 2}, ∀t ∈ [O, T ], φ(i)(., t) : Ω(i)

o 3 X(i)7→ x(i)= φ(i)(X(i), t) (1)

where X(i)and x(i)_{are the reference and current positions of any particle of body i, respectively.}

The displacement field in body i is defined as U(i)_{= x}(i)_{− X}(i)_{. The strong form of the problem is expressed by the}

local equations summarized below, which must be satisfied at any time t ∈ [O, T ] and at any point X(i)∈ Ω(i)o , i ∈ {1, 2}.

The linear momentum balance equation can be written for each body i via divΠ(i)_{+ ρ}(i)

o f(i)= ρ(i)o U¨ (i)

(2) where Π(i)is the first Piola-Kirchhoff stress tensor, ρ(i)o is the reference density of body i and f(i)the prescribed body force

per unit mass.

The constitutive relationships relating stress to strain are left unspecified until the numerical examples are considered. This means that the weighted residual relationship proposed hereafter holds for all constitutive materials, as is the case with the standard virtual work principle.

The reference boundary So(i)of body i is partitioned into three non-overlapping and pairwise disjoint parts denoted SoU(i),

S_oT(i)and Soc(i), where S_oU(i) and S_oT(i)are the parts where the displacement and the tractions are prescribed, respectively, and

Soc(i)is the part where contact potentially takes place between the two bodies during some portion of the time interval [O, T ].

The boundary conditions on S_oU(i)and S_oT(i)on each body i can be summarized as U(i)_{= ¯U}(i)

in S_oU(i), Π(i)_N(i)_{= ¯T}(i)

in S_oT(i) (3)

where ¯U(i), ¯T(i)are the prescribed displacement and nominal traction, respectively, N(i)the unit outward normal at point X(i)to surface So(i). The boundary conditions on the contact surface will be presented in detail in Section 2.3. The initial

conditions for body i are

∀X(i)∈ Ω(i)o , U(i)(X(i), 0) = 0 ˙U

(i)

(X(i), 0) = ¯V(i)0 (X(i)) (4)

2.1 Contact kinematics

Let us now focus on the equations governing the response on the contact interface, which are the essential ingredients in the large deformation contact problem. Let the spatial counterparts of Ωo(i), So(i), S(i)_oU, S_oT(i)and Soc(i)be denoted by Ω(i), S(i),

S_U(i), S_T(i)and Sc(i), respectively, e.g., S(i)c = φ(i)(Soc(i), t). In order to describe the relative motion of the two bodies, we

arbitrarily choose one body (the target), say body 2, as the reference and to evaluate the proximity of the other (the contactor), thus body 1, with respect to it.

It is assumed that the reference surface Soc(i), i = 1 or 2, is parametrized by the mapping θ(i)and the current surface Sc(i)

at time t by the mapping ψ(i)(., t) = φ(i)(., t) ◦ θ(i):

A region of R2 _{→ S}(i)

oc ⊂ R3 → Sc(i)⊂ R3

ξ(i)_{= (ξ}(i)1_{, ξ}(i)2_{) 7→ X}(i)_{= θ}(i)_(ξ(i)_{) 7→ x}(i)_{= φ}(i)_(X(i)_{, t) = ψ}(i)_(ξ(i)_{, t)} (5)

Consider a particular point x ∈ Sc(1) and define the opposite point y ∈ Sc(2)as the closest point to x in the Euclidean

sense (Figure 1): y = arg min

x(2)_∈S(2)

c

kx − x(2)_{k (6)}

In writing (6), we assume that point y is unique without discussing the case of non-uniqueness, which is not of major consequence in numerical computations. The interested readers may consult comprehensive comments on this issue in the literature, e.g., [7], [2] p48, [3] p115, and references quoted therein. If the target surface Sc(2) is smooth at y, then y is the

projection of x on Sc(2). As may be seen, the notation x(i), i ∈ {1, 2}, in (5) designates any point on Sc(i), whereas the

notation x designates a particular point of interest on S(1)c and y the opposite point of x on Sc(2)defined by (6).

Y

X

x

y

)

(1

oc

S

S

S

S

η

ξ

ν

θθθθ

θθθθ

Ψ

Ψ

Ψ

Ψ

(.,t)

φφφφ

(.,t)

φφφφ

(.,t)

Ψ

Ψ

Ψ

Ψ

(.,t)

Figure 1: Contact kinematics.

The point x ∈ Sc(1)being associated to (at least) one point y ∈ Sc(2), the proximity g is defined by

g = −ν(x − y) (7)

where ν is the normal at point y. The sign convention chosen in (7) implies that the proximity g is positive when there is interpenetration and negative when there is a gap between the two bodies. Thus, symbol g is the opposite of what is referred to as the gap in the literature. It can be checked that g is an objective quantity.

Let us now convert all the spatial quantities defined above into material ones defined on the reference configurations, by means of the motions φ(1) and φ(2) in (1). Considering the point X ∈ Soc(1) related to the point x considered by x =

φ(1)(X, t), then the point Y ∈ Soc(2)related to point y by y = φ(2)(Y, t) satisfies

Y = arg min

X(2)_∈S(2)

oc

kφ(1)(X, t) − φ(2)(X(2), t)k (8)

A point Y thus being associated to point X, the proximity g defined in (7) can be rewritten as

Eventually, the inverse image of X (resp. Y) under the parametrization θ(1)(resp. θ(2)) will be denoted ξ (resp. η). By making x(2)_{= ψ}(2)_(ξ(2)_{, t) in (6), one finds that η is characterized by}

η = arg min

ξ(2) kx − ψ

(2)_(ξ(2)_{, t)k (10)}

For notational conveniences, the dependencies on φ(1), φ(2)and ψ(2)will be omitted in the material kinematics variables and only the dependency on (X, t) will be shown. Thus, for instance, one writes η = η(X, t) and g = g(X, t).

Assuming that all the mappings introduced are smooth enough, let us define the natural basis (a1, a2) at point y ∈ Sc(2)

as a1= ∂ψ (2) ∂ξ(2)1(η(X, t), t) a2= ∂ψ(2) ∂ξ(2)2(η(X, t), t) (11)

The surface Sc(2)is parametrized conveniently so that the normal n = a1× a2/ka1× a2k at y to Sc(2)is directed toward

the contactor. Use will also be made of the dual basis (a1_{, a}2_{) lying in the tangent plane of the current target surface as}

follows

∀α ∈ {1, 2}, aα= aαβaβ (12)

where implicit summation from 1 to 2 is assumed over repeated Greek indices and the 2 × 2 matrix [aαβ_{] is the inverse of}

matrix [aαβ] = [aα.aβ].

The contact kinematics with friction is described here by using the following objective relative velocity at point X and time t [19, 20]

V = V(1)(X, t) − V(2)(Y(X, t), t) + g ˙ν = − ˙gν + ˙ηα_a

α (13)

where V(1)(X, t) and V(2)(Y, t) are the velocities of the particles X and Y(X, t), respectively. The rates ˙g, ˙ηα _{and ˙ν are}

given by [2, 3]: ˙g = −hV(1)(X, t) − V(2)(Y(X, t), t)iν ˙ηα_{= ¯a}αβ_a β[V(1)(X, t) − V(2)(Y(X, t), t)] − g¯aαβνV(2),β (η, t) ˙ν = −(καγ¯aγβaα⊗ aβ)(V(1)− V(2)) − (¯aαβaβ⊗ ν)V(2),α (14)

where matrix [¯aαβ_{] is the inverse of matrix [¯a}

αβ], with ¯aαβ = aαβ+ gκαβ, καβ = νψ,(2)αβ is the curvature of the surface

ψ(2)at η, the symbol ⊗ designates the tensor product defined as (a ⊗ b).c = a(b.c) for all vectors a, b and c. Note that tensor καγ¯aγβaα⊗ aβ= ¯aαγκγβaα⊗ aβin (14c) is symmetric.

On the basis of (13) one can define (i) the relative normal velocity VN as the projection of V on the normal vector ν, and

(ii) the slip velocity or the relative tangential velocity VT as the projection of V into the tangent plane at y to Soc(2):

VN = V.ν = − ˙gν VT = (I − ν ⊗ ν).V = ˙ηαaα (15)

It should be noted that the slip rate VT associated to the material point X ∈ Soc(1) is resolved in terms of the spatial

basis (a1, a2) at point y ∈ Sc(2). As done with other kinematical quantities, use will be made of the abbreviated notations

V = V(X, t) and VT = VT(X, t), omitting the dependencies on motions φ(1), φ(2).

2.2 Contact tractions

Let the contact Cauchy traction vector t(x, t) at a typical point x ∈ Sc(1)be resolved in terms of the local basis (a1, a2, ν) at

the projection point y ∈ Sc(2)as follows

t = tNν − tT = tNν − tT αaα (16)

The normal component tNof traction vector t in the direction of normal ν is the contact pressure, positive in compression.

The tangential component tT = tT αaαis the opposite of the frictional traction. It can be verified that the pressure tN and

the frictional traction tT are objective variables.

Let T(X, t) = Π(1).N be the nominal Piola-Kirchhoff traction vector at point X ∈ Soc(1) (N the normal vector at X to

Soc(1)). Like the contact Cauchy traction, the nominal traction T is resolved in terms of the spatial basis at y ∈ Sc(2)as follows

T(X, t) = TN(X, t)ν − TT(X, t) = TN(X, t)ν − TT αaα (17)

From relation tdS(1) _{= TdS}(1)

o (dSo(1) is a differential reference area, dS(1) its spatial counterpart), one derives the

following relationships between the spatial and material traction components

2.3 Contact laws

As a means of describing the interfacial response at current time t, the contact laws are naturally expressed in the spatial form. However, one may use (18) in order to recast them into the mixed form involving the relative material velocity and the nominal contact tractions, which are more suitable for the numerical implementation within the Lagrangian description framework. The normal contact law then reads

∀t ∈ [O, T ], ∀X ∈ Soc(1), g(X, t) ≤ 0

½

. if g(X, t) < 0, then TN(X, t) = 0

. if g(X, t) = 0, then TN(X, t) ≥ 0 (19)

In this work, the tangential contact response is modeled by the Coulomb friction law:

∀t ∈ [O, T ], ∀X ∈ Soc(1),

(a) g(X, t) < 0 ⇒ TT(X, t) = 0 (no relationship for VT(X, t))

(b) g(X, t) = 0 ⇒ kTT(X, t)k ≤ µTN(X, t) where

½

. if kTTk < µTN, then VT = 0 (stick)

. if kTTk = µTN, then VT × TT = 0, VT.TT ≥ 0 (slip)

(20)

where µ > 0 is the coefficient of friction.

It can be verified that the contact laws (19)-(20) are objective.

3 Weighted residual relationship for the contact problem

The strong form of the contact problem is described by equations (1)-(2), the constitutive law, the boundary conditions (3), the contact laws (19)-(20) and the initial conditions (4). In this section, we convert the strong form into an equivalent weak one which is more convenient for the finite element implementation.

The approach chosen is inspired by the augmented Lagrangian formulation of Pietrzak and Curnier [13] and amounts to rewrite the augmented Lagrangian formulation of Alart and Curnier [4] in a generic weak form. More specifically, a weighted residual relationship is proposed which involves the displacement fields U(i), i ∈ {1, 2}, defined in Ω(i)o , and the

multiplier field λN(X) and λT(X) = λT αaαdefined on Soc(1). Accordingly, the weighting functions are the virtual

displace-ments U(1)∗, U(2)∗, and the virtual multipliers λ∗

N, λ∗T = λ∗T αaα. Two positive constants ²N, ²T being chosen, the weighted

residual relationship is stated in the following proposition [21]. PROPOSITION1.

I The solution fields of the strong problem – namely U(1), U(2) in Ω(1)o , Ω(2)o and TN, TT, VT on Soc(1) – satisfy the

following relationship, provided one makes λN = TN and λT = TT:

∀t ∈ [O, T ], ∀U(1)∗_{, ∀U}(2)∗_{, ∀λ}∗

N, ∀λ∗T 1, ∀λ∗T 2, 2 X i=1 ( − Z Ω(i)o

Π(i)T : ∇_X(i)U(i)∗dΩo+

Z Ω(i)o ρ(i) o f(i)U(i)∗dΩo+ Z S_oU(i)∪S(i)_oT T(i)U(i)∗dSo ) + Z Soc(1) · hλN + ²Ngiν − µ 1 − h1 − µhλN+ ²Ngi kλT + ²TVTki ¶ (λT + ²TVT) ¸ ³ U(1)∗(X) − U(2)∗(Y(X)) ´ dSo + Z Soc(1) ½ (λN− hλN + ²Ngi) λ∗ N ²N + · λT − µ 1 − h1 − µhλN+ ²Ngi kλT + ²TVTki ¶ (λT + ²TVT) ¸ λ∗ T ²T ¾ dSo = 2 X i=1 Z Ω(i)o ρ(i)o U¨ (i) U(i)∗dΩo (21)

where ∇_X(i)U(i)∗is the gradient tensor of U(i)∗with respect to variables X(i), h.i is the Macauley bracket: hai = a if a ≥ 0, = 0 if a < 0, and h1 − µhλN+²Ngi

kλT+²TVTki must be replaced by 0 at any point on S

(1)

oc where λT + ²TVT = 0.

I Conversely, the contact surfaces Sc(1)and Sc(2)being parametrized by ξ 7→ x ∈ Sc(1)and η 7→ y ∈ Sc(2), it is assumed

that there is a bijective (or piecewise bijective) mapping Sc(1) 3 x 7→ y ∈ Sc(2)between them. Then, relation (21) implies at

any time t ∈ [O, T ] the following local equations:

(1) The momentum balance equation (2) for the two bodies 1 and 2.

(2) Relation (3b) on the boundary portion S(i)o \Soc(i)= S(i)_oT∪ S_oU(i): Π(i)N(i)= T(i).

(3) The following equalities between the components of the nominal traction vectors and the multipliers on the contactor surface Soc(1)

∀X ∈ S(1)

(4) The normal and tangential contact laws (19) and (20).

(5) The following relationship which expresses the equilibrium of the traction vectors at the contact interface

∀X ∈ S(1) oc , T(2)(Y)dSoc(2)= −(λNν − λT)kX,ξ 1×X,_ξ2k kY,η1×Y,_η2k Dξ DηdS (2) oc = −T(1)(X)dSoc(1) (23)

where point Y = Y(X, t) ∈ Soc(2) corresponds to point X through the above-mentioned bijection, dSoc(1) is a differential

reference area in Soc(1)and dSoc(2)its counterpart in Soc(2)related to dS(1)oc.

REMARKS.

(i) In general, λT+ ²TVT differs from zero since λT and VTdo not vanish simultaneously: if the slip rate vanishes, then

the tangential traction does not, and vice versa. However, in some problems with a specific symmetry, there may be some points where λT + ²TVT is zero. In practice, the set of these points is either isolated points or lines on the contact surface

Soc(1).

(ii) In dynamics, when shocks and impacts are expected, the acceleration ¨U(i)should be understood as the derivative of the velocity in the sense of the distributions [22] ¥

Proof.

The first part of the above proposition, which states that the strong form implies the weak one, can be readily checked. Let us prove the reciprocal statement for an arbitrary fixed time t ∈ [O, T ].

• Since (21) holds for all virtual fields U(1)∗, U(2)∗, λ∗

N and λ∗T, let us pick U(1)∗ = U(2)∗ = 0 and λ∗T = 0, while

keeping λ∗

N arbitrary. Relationship (21) then leads to

λN = hλN+ ²Ngi on Soc(1) (24)

The first consequence of (24) is λN ≥ 0. Next, two mutually exclusive cases arise:

. either λN + ²Ng < 0, then (24) entails

λN = 0 ⇒ g < 0 (since λN+ ²Ng < 0) (25)

. or λN+ ²Ng ≥ 0, then (24) entails

λN = λN+ ²Ng ⇔ g = 0, which gives λN ≥ 0 since λN+ ²Ng ≥ 0 (26)

The results for the two cases above can be gathered together as follows

g ≤ 0 where

½

. if g < 0, then λN = 0

. if g = 0, then λN ≥ 0 (27)

which is the normal contact law (19) provided λN = TN which will be shown later.

• Consider (21) again and choose now U(1)∗ = U(2)∗ = 0 and λ∗

N = 0, whereas λ∗T is left arbitrary. The localization

theorem can then be applied assuming that the coefficient of λ∗T is a continuous function in space. At any point on Soc(1)

where λT + ²TVT 6= 0, one has

λT = µ 1 − h1 − µhλN + ²Ngi kλT + ²TVTki ¶ (λT+ ²TVT) (28)

The first consequence of (28) is that the slip velocity VT is parallel to λT. Let us show a more precise result, specifically

VT is oriented in the same direction as λT. To do this, let us temporarily denote α = _kλµhλ_TN_+²+²_TN_VTgi_k ≥ 0 for brevity. Relation

(28) then reads

h1 − αiλT = (1 − h1 − αi)²TVT (29)

As α ≥ 0 and 1 − h1 − αi ≥ 0, ∀α, one has the following mutually exclusive cases depending on α’s value: 

 

. if α = 0, then (29) gives λT = 0.

. if 0 < α < 1, then 1 − α > 0, 1 − h1 − αi > 0 and (29) entails (1 − α)λT = α²TVT.

. if α ≥ 1, then (29) gives VT = 0.

(30) Thus, in any case VT is oriented in the direction of λT. Now, let us distinguish two cases:

- If λN+ ²Ng < 0, then one gets from (25) and (28) g < 0 and λT = 0.

- If λN+ ²Ng ≥ 0, then one gets from (26) g = 0. Two mutually exclusive sub-cases arise here:

. if kλT + ²TVTk ≥ µ(λN + ²Ng), then (28) entails λT = µ(λN + ²Ng)_kλλT_T+²_+²T_TV_VT_Tk.

By taking the norm of both sides of the last equality, recalling here λN+ ²Ng ≥ 0 and g = 0, one finds kλTk = µλN.

The last results can be summarized as follows:    . if g < 0, then λT = 0 . if g = 0, then kλTk ≤ µλN, and ½ . if kλTk < µλN, then VT = 0

. if kλTk = µλN, then VT is oriented in the direction of λT

(31) which is merely the tangential contact law (20) provided λN = TN, λT = TT, which will be shown soon.

• Eventually, let us make in (21) λ∗

N = 0, λ∗T = 0 and keep U(1)∗ and U(2)∗ arbitrary. According to the divergence

theorem, relationship (21) leads to the following equality

2 X i=1 { Z Ω(i)o

U(i)∗(divΠ(i)+ ρ(i)o f(i))dΩo−

Z

S(i)_oU∪S(i)_oT

U(i)∗.(Π(i).N(i)− T(i))dSo}

− Z Soc(1) U(1)∗_.Π(1)_.N(1)_dS o+ Z Soc(1) U(1)∗ · hλN+ ²Ngiν − µ 1 − h1 − µhλN+ ²Ngi kλT+ ²TVTk i ¶ (λT + ²TVT) ¸ dSo − Z Soc(2) U(2)∗_.Π(2)_.N(2)_dS o− Z Soc(1) U(2)∗ · hλN+ ²Ngiν − µ 1 − h1 − µhλN+ ²Ngi kλT+ ²TVTk i ¶ (λT + ²TVT) ¸ dSo = 2 X i=1 Z Ω(i)o ρ(i)o U¨ (i) U(i)∗dΩo (32)

where N(i)is the unit outward normal at point X(i)∈ So(i).

Note that the last integral of the left-hand side of (32) is taken over Soc(1). It can be recast into an integral over S(2)oc by

using the chain of bijections ξ 7→ X 7→ Y 7→ η which allows one to change the integration variables

dS(1) o = kX,ξ1×X,_ξ2kdξ1dξ2= kX,_ξ1×X,_ξ2kDξ Dηdη 1_dη2 _dS(2) o = kY,η1×Y,_η2kdη1dη2 (33) Hence dSo(1)= kX,ξ1×X,_ξ2k kY,η1×Y,_η2k Dξ DηdS (2) o (34)

By substituting (34) into (32), one gets relations (2), (3b), (23) and

T(1)= Π(1)N(1)= hλN+ ²Ngiν − µ 1 − h1 − µhλN + ²Ngi kλT+ ²TVTki ¶ (λT + ²TVT) sur S(1)oc (35)

Comparing (35) with the definition (17), T(1) = TNν − TT, and making use of (24) and (28) lead to equality (22)

between the contact tractions and the multipliers. Next, the normal and tangential contact laws (19) and (20) are derived from (22), (27) and (31).

• It remains to consider the points on Soc(1)where λT+²TVT = 0. Equality (28) is then replaced by λT = λT+²TVT,

which entails λT = VT = 0. The subsequent reasoning is analogous yet simpler than the case λT + ²TVT 6= 0. It can

easily be checked that the tangential contact law (20) as well as (Relations 22) and (23) are satisfied at the points where

λT + ²TVT = 0. The whole Proposition 1 has been proved ¥

4 Semi-discrete equation system

In this section, the weighted residual relationship (21) will be discretized in space by means of the finite element method. The discretization process will be described in the 3D framework only but the results for 2D problems can be derived in a similar manner. The following shorthand notations will be used for brevity

ˆ

λN = λN+ ²Ng λˆT = λT+ ²TVT (36)

We shall confine ourselves to the terms pertaining to contact in equality (21), denoted W∗

contact, since the other terms can

be discretized in a standard way. Let the contactor surface Soc(1)be subdivided into a number of isoparametric finite elements

e(1)_{(the superscript ‘(1)’ is used throughout to remind of the contactor surface S}(1)

as the sum of the element contact terms (W∗ contact)e

(1)

related to elements e(1) _{and evaluated using the Gauss quadrature}

rule: (W∗ contact)e (1) = X quadrature points (" hˆλNiν − Ã 1 − h1 −µhˆλNi kˆλTk i ! ˆ λT # ³ U(1)∗(X) − U(2)∗(Y(X)) ´ +³λN − hˆλNi´ λ ∗ N ²N + " λT − Ã 1 − h1 −µhˆλNi kˆλTk i ! ˆ λT # λ∗ T ²T ) j(ξ) (37)

where all terms are computed at Gauss points and j(ξ) is the area jacobian times the weight corresponding to the parent coordinates ξ = (ξ1, ξ2) ∈ R2of quadrature point X ∈ e(1), see Figure 1

j(ξ) = kX,ξ1(ξ) × X,_ξ2(ξ)k.(weight at location ξ) (38)

In the sequel, the following notational convention is adopted for the matrix representations: curly brackets { } designate a column vector, the angle brackets < > an arrow vector, and the square brackets [ ] a general matrix. In any element

e(1)_{⊂ S}(1)

oc , the reference coordinates X, the displacement U(1)and the multipliers λ are interpolated by

{X} = [N(1)_(ξ)]e(1) {X}e(1) {U(1)} = [N(1)_(ξ)]e(1) {U(1)}e(1) {λ} = [N(1)_(ξ)]e(1) {λ}e(1) (39)

In (39b) for instance, {U(1)_{} is the column vector with the three components of the displacement U}(1)_{related to a fixed}

Cartesian orthonormal basis (e1, e2, e3). The column vector {U(1)}e

(1)

contains the 3nne(1)_{nodal displacement components}

associated with element e(1)_{, where nne}(1) _{is the number of nodes of element e}(1)_{. The matrix [N}(1)_(ξ)]e(1)

contains the shape functions of element e(1)_{and is of dimension 3 × 3nne}(1)_.

The normal and tangential components of multiplier λ can be derived from (39c) via

λN =< ν > [N(1)(ξ)]e (1) {λ}e(1) {λT} = − [I − ν ⊗ ν] [N(1)(ξ)]e (1) {λ}e(1) (40)

where {λT} is a 3-component column vector.

Like the contactor surface, the target surface Soc(2)is subdivided into a number of isoparametric finite elements. Given a

quadrature point X in element e(1)_{, let e}(2)_{the (or an) element of S}(2)

oc containing the reference point Y corresponding to the

projection y = proj_S(2)

c x ∈ S

(2)

c . In element e(2), the reference coordinates Y and the displacement U(2)are interpolated by

{Y} = [N(2)_(η)]e(2)

{Y}e(2)

{U(2)} = [N(2)_(η)]e(2)

{U(2)}e(2) (41)

where η = (η1, η2) ∈ R2is the parent coordinate of Y and the superscript (2) is used to remind of element e(2) ⊂ Soc(2). The

column vector {U(2)}e(2)

contains the 3nne(2)_{nodal displacement components associated with element e}(2)_{, where nne}(2)

is the number of nodes of element e(2)_{. The 3 × 3nne}(2)_{matrix [N}(2)_(η)]e(2)

contains the shape functions of element e(2)_.

As in [3], given a quadrature point X in element e(1) _{⊂ S}(1)

oc , there is at least one element e(2) ⊂ Soc(2) containing the

reference point Y of y = proj_S(2)

c x. Both element e

(2)_{and the number of nodes nne}(2)_{, which depend on the projection y,}

vary as a function of the quadrature point X. Moreover, the whole set of quadrature points of element e(1)_{generally involves}

several elements e(2)_{in S}(2)

oc .

All the virtual quantities are interpolated in a similar way. Eventually, the element contact expression (W∗ contact)

e(1) in (37) on the element e(1)_{level can be written in the discretized form}

(Wcontact∗ )e (1) = X quadrature points ³ << U(1)∗>e(1), < U(2)∗ >e(2)> {Φcontact}e− < Λ∗>e (1) {RΛ}e ´ j(ξ) ₍₄₂₎

where {Φcontact}eis the element contact force vector and {RΛ}ethe element residual vector for the multipliers, defined as

{Φcontact}e=            [N(1)_(ξ)]e(1)_T Ã hˆλNi{ν} − Ã 1 − h1 − µhˆλNi kˆλTk i ! {ˆλT} ! −[N(2)_(η)]e(2)_T Ã hˆλNi{ν} − Ã 1 − h1 −µhˆλNi kˆλTk i ! {ˆλT} !            (43)

{RΛ}e= [N(1)(ξ)]e (1)_T Ã 1 ²N ³ hˆλNi − λN ´ {ν} + 1 ²T ( λT− Ã 1 − h1 −µhˆλNi kˆλTk i ! ˆ λT )! (44) Vector {Φcontact}e in (43) has nne(1) + nne(2) components, where nne(2) varies as a function of the considered

quadrature point; vector {RΛ}ehas nne(1)components. Note that

1 − h1 −µhˆλNi kˆλTk i =      0 if ˆλN < 0 (algorithmic gap) µˆλN

kˆλTk if ˆλN ≥ 0 and kˆλTk ≥ µˆλN (algorithmic contact with slip)

1 if ˆλN ≥ 0 and kˆλTk < µˆλN (algorithmic contact with stick)

(45)

In (45), use has been made of the adjective ‘algorithmic’ in order to emphasize the difference with the ‘real’ situation. For instance, the ‘algorithmic gap’ case means ˆλN = λN + ²Ng < 0, in contrast with the real ‘gap’ case which merely

means g < 0; and the ‘algorithmic slip’ case means kˆλTk ≥ µˆλN, as opposed to the ‘slip’ case which means kλTk = µλN.

By adding the element contributions in (42), one obtains the following semi-discrete coupled equation system [M]{ ¨U} + {Ψ(U)} = {Φ(U)} + {Φcontact(U, Λ)}

{RΛ(U, Λ)} = {0} (46)

where vectors {Φcontact(U, Λ)} and {RΛ(U, Λ)} are the assembly of the element vectors {Φcontact}ej(ξ) and {RΛ}ej(ξ)

given in (43) and (44), respectively. The unknowns in (46) are the nodal displacement vector {U} = ½

{U(1)_}

{U(2)}

¾ in the two bodies 1 and 2, and the multiplier vector {Λ} on the reference contact surface Soc(1). Except for the above-mentioned

vectors {Φcontact(U, Λ)} and {RΛ(U, Λ)} which are specific to contact, the other vectors and matrices are standard and

result from the discretization of terms other than W∗

contactin relationship (21): [M] is the mass matrix, {Ψ(U)} the internal

force vector resulting from the strain work in the bodies, {Φ(U)} the external force vector (excluding contact force) which may depend on displacement {U} if follower loads (e.g., a pressure) are applied on either body. The nonlinear semi-discrete equations (46) will be solved as a whole using the standard full Newton-Raphson method to simultaneously obtain the nodal displacements and the multipliers, as was done in [4, 7, 13, 8] (Curnier and co-workers even used a so-called generalized Newton method which is more sophisticated).

• The slip velocity VT = ˙ηαaα defined in (15) and involved in (43)-(44) has to be computed here by means of an

adequate temporally discrete expression. For this purpose, let us first note that Proposition 1 remains valid if VT is replaced

with VT∆t, where ∆t is an arbitrary finite time interval. The quantity VT∆t having the dimension of a length will be

referred to as the slip, its use instead of the slip velocity is more consistent in the numerical context because the penalty parameters ²N and ²T involved in the weak form then have the same dimension of a force by unit volume, i.e. the same

SI units of N/m3_{. Let us now look at a typical iteration of the current time step n and introduce some new notations as}

summarized in Figure 2. For the sake of brevity, any quantity computed at the current step n will not be indexed by n, whereas any quantity related to the previous step will be distinguished by subscript n − 1. Consider two times t and tn−1,

which are respectively the current time corresponding to the current iteration of the current step n and the time corresponding to the previous step (n − 1). Given a particle on the contactor surface Sc(1)with reference position X, its position at time

tn−1is xn−1= φ(1)(X, tn−1) and its current position is x = φ(1)(X, t). We denote the following quantities related to time

tn−1:

. y_n−1the projection of xn−1on the target surface Sc(2)(tn−1) at time tn−1,

. Yn−1and ηn−1the inverse images of yn−1under the chain of parametrizations (5).

It should be emphasized that (i) y and y_n−1are not the respective positions at times t and tn−1of a same particle in body

2 and (ii) the current position of the particle Yn−1is φ(2)(Yn−1, t), in general distinct from y = φ(2)(Y, t).

With these notations in hand, using the chain of parametrizations (5), which gives y = ψ(2)(η, t) and ψ(2)(ηn−1, t) =

Φ(2)(Yn−1, t), and applying the backward Euler integration scheme with time step ∆t = t − tn−1leads to the following

expression for the slip

VT∆t = y − Φ(2)(Yn−1, t) (47)

It should be noted that other expressions such as VT∆t = ψ(2)(η, t) − ψ(2)(ηn−1, t) would not be suitable as surface

Sc(2)is divided into finite elements and the parametrization (5) is in fact defined piecewisely. It may happen that the projection

points yn−1and y do not belong to the same element on Sc(2)although they are close to each other; in this case their local

parent coordinates ηn−1and η may be quite distant even though the time step ∆t tends to zero. The algebraic expression

(47) does not involve the parent coordinates and thus avoids the aforementioned discontinuity problem. It is adopted to compute the slip in the contact force vector (43) and the residual (44), as well as the tangent stiffness in the next section.

y

x

x

y

Φ

_(Y

,t)

Configuration at previous step

(time t

)

)

(

t

S

₍

₎

t

S

)

(

t

S

V

∆t

)

(

t

S

Current configuration

(time t)

• The important issues of convergence and stability of a mixed finite element formulation were investigated in the

literature for continuum problems and later for contact problems. They are related to the inf-sup condition which is expressed in contact problems as a compatibility condition between the displacement and multiplier spaces on the contact surfaces. Following Chapelle and Bathe’s developments [23] on incompressible elasticity, El-Abbasi and Bathe [24] examined the small deformation frictionless contact problem and proposed an inf-sup test procedure where a series of refined meshes is used in order to evaluate the evolution of the numerical inf-sup value. A further mathematical study was carried out by Bathe and Brezzi in [25], giving rise to precise relations between the degrees of the polynomial interpolations for the displacements and the multipliers. In general, the mathematical study of the inf-sup condition is confined to the small deformations framework since it is difficult when the structure undergoes large deformations and the contact surface is unknown a priori. No attempt has been made in this work in order to prove - analytically or numerically - that the proposed method passes the inf-sup condition. This issue should be investigated in the future. In the meanwhile, it can be observed through the numerical examples presented in Section 6 that the convergence is achieved when the mesh is refined and there is neither locking nor spurious contact pressure.

5 Contact tangent stiffness matrix

This section is devoted to the derivation of the tangent stiffness matrix required for the Newton-Raphson solution of the coupled equation system (46). We will confine ourselves to that part of the tangent stiffness related to contact, since the remaining part related to non contact terms can be linearized in a standard way. Let δU and δλ be the independent variations of the displacement and the multiplier, respectively defined by

δU = δUi_e

i, δλ = δλNν − δλT, δλT = δλT αaα⊥ ν (48)

where Latin indices take the values 1, 2, 3, Greek indices take the values 1, 2, repeated indices imply summation and the variations δUi_{, i ∈ {1, 2, 3}, δλ}

N, δλT 1, δλT 2are arbitrary. One has to compute the exact linearization of Wcontact∗

corresponding to these variations and extract the contact stiffness [Kcontact] from the relation

∀{δU}, {δΛ}, δW∗ contact= − ½ {U∗_} {Λ∗_} ¾T [Kcontact] ½ {δU} {δΛ} ¾ (49) The variation (δW∗ contact)e (1)

over any element e(1) _{⊂ S}(1)

oc is obtained by taking the directional derivatives of (37) in

the directions δU(i)_{, i ∈ {1, 2}, δλ}

N and δλT. The variations of the basic kinematic variables are expressed in terms of the

displacement variations via [2, 3]

δg = −(δU(1)_{(X) − δU}(2)_{(Y(X, t)).ν}

δηα_{= ¯a}αβ_a

β(δU(1)(X) − δU(2)(Y(X, t)) − g¯aαβνδU(2)_,β (η)

δν = −(καγ¯aγβaα⊗ aβ)(δU(1)(X) − δU(2)(Y(X, t)) − (¯aαβaβ⊗ ν)δU(2),α(η)

where δU(2),α(η) denotes the partial derivative with respect to ηαof the composite function δU(2)(η) = δU(2)(Y(η)).

As for the variation of the slip, it is derived from (47):

δ(VT∆t) = δU(2)(Y) − δU(2)(Yn−1) + W.(δU(1)(X) − δU(2)(Y)) − ZαδU(2),α(η) (51)

where W and Zα, α ∈ {1, 2}, are non symmetric second-order tensors defined as W = ¯aαβ_a

α⊗ aβ Zα= g¯aαβaβ⊗ ν (52)

Like the displacement, the variations of the displacement and the multiplier are obtained by interpolating the nodal variations

{δU(1)} = [N(1)(ξ)]e(1){δU(1)}e(1) {δU(2)} = [N(2)(η)]e(2){δU(2)}e(2) {δλ} = [N(1)(ξ)]e(1){δλ}e(1) (53)

In the above relationship, {δλ} for instance is the 3-component column vector representing δλ in the fixed basis (e1, e2, e3) and the column vector {δλ}e

(1)

with 3nne(1)_{components contains the triplets of the nodal components of δλ in}

the same basis. One also needs to interpolate the variation {δU(2)(Yn−1)} involved in (51)

{δU(2)_(Y n−1)} = [N(2)(ηn−1)]e (2) n−1{δU(2)}e (2) n−1 (54)

where e(2)_n−1denotes the element in the reference target surface S(2)oc containing point Yn−1, in general different from element

e(2)_.

Eventually, relations (50)-(54) enable one to express the element linearization (δW∗ contact) e(1) in the form (δWcontact∗ )e (1) = − X quadrature points      {U(1)∗}e(1) {U(2)∗}e(2) {Λ∗}e(1)      T [Kcontact]ej(ξ)          {δU(1)}e(1) {δU(2)}e(2) {δU(2)_}e(2)_n−1 {δΛ}e(1)          (55)

where the element contact stiffness [Kcontact]eis a rectangular matrix of the form

[Kcontact]e= · [KU U]e [KU Λ]e [KΛU]e [KΛΛ]e ¸ (56) The explicit expression of the above matrix is given in the Appendix. It should be emphasized that (i) despite the superscript e which stands for element, [Kcontact]eis actually defined for each quadrature point of element e(1) ⊂ Soc(1)and

(ii) although the element matrices [Kcontact]ej(ξ) are rectangular, their assembly eventually results in the square contact

stiffness [Kcontact] of the form

[Kcontact] = · [KU U] [KU Λ] [KΛU] [KΛΛ] ¸ (57)

6 Numerical examples

This section presents the numerical examples which are carried out on the basis of the previous finite element formulation in order to assess the efficiency of the proposed formulation. Although the semi-discrete equation system (46) holds in dynamics, from now on we will confine ourselves to those quasistatic cases that display enough specific features of the contact phenomenon to consider the formulation validated. The dynamic cases present additional physical and numerical difficulties and are deliberately disregarded at this stage. All numerical results presented here have been obtained by a home made finite element code.

6.1 Contact patch test

The first example we consider is related to the contact patch test which was originally conceived in [26] and [27], with the aim of checking whether the finite element contact formulation allows transmission of a constant pressure through an arbitrary nonconforming contact surface.

In this work, we used the patch test proposed in [28, 24] and shown in Figure 3a. An elastic body is pressed against another one by a uniform distributed load p = 1000N/m2 _{applied on its top surface. Assuming a frictionless interface}

contact line. The patch test is passed if the numerical results yield such a uniform stress distribution, regardless of the mesh used. In practice, one has to verify the test using some typical meshes.

Here the problem is solved assuming that the two bodies are made of an isotropic hyperelastic Saint-Venant Kirchhoff material characterized by the strain energy function in terms of the Green strain E

w(E) = 1

2λ(trE)

2_{+ ¯µtr(E}2_{) (58)}

where the Lamé constants λ, ¯µ are related to Young’s modulus E and Poisson’s ratio ν by the same relations as in linear

elasticity

λ = Eν/(1 + ν)/(1 − 2ν) µ = E/2/(1 + ν)¯ (59)

The problem is modeled in plane strain conditions, taking the dimensions of the bodies a = 5m and b = 1m, and the material properties E = 106_N/m2_{and ν = 0. The structure is discretized using two meshes – the first one is regular and the}

second distorted – as shown in Figure 3b-c, in both cases the mesh is nonconforming along the contact interface. The two bodies are discretized with 8-node quadrilaterals (Q8), so that the elements on the contact surface are 3-node line elements. The numerical quadrature is performed using 3 × 3 Gauss quadrature points in the Q8 elements and 7 Gauss points in line elements belonging to the contact surface. The penalty parameter is ²N = 1012N/m3.

p b b E, ν y E, ν a (a) (b) (c)

Figure 3: Contact patch test. (a) Problem definition, (b) Regular mesh, (c) Distorted mesh.

Figure 4 shows the deformed configurations and contact stress distributions along the contact line when the upper, respectively the lower, body is selected to be the contactor. The arrows represent the contact stresses exerted by the target body upon the contactor. In all cases, the contact stresses are found to be constant and equal to p up to 5 or 6 significant digits.

All the nodes on the contact interface have the same vertical displacement Uy = −0.0010015m. The stress field inside

the two bodies is constant, with the only non-zero stress component Σyy = −1001.N/m2(Σ is the second Piola-Kirchhoff

stress tensor). These values agree with the analytical solution of the nonlinear elasticity (nonlinear elasticity must be used for the comparison since the numerical results are obtained by a nonlinear finite element code).

6.2 Hertzian contact problem

In this example, we consider a Hertzian contact problem in order to investigate the accuracy of the proposed contact formu-lation. Referring to Figure 5a, a 2D elastic cylinder with radius R = 10m is pressed against a rigid foundation by a force F which is perpendicular to the rigid plane and applied on the symmetry axis of the cylinder. The cylinder is made of the same isotropic hyperelastic Saint-Venant Kirchhoff material as in (58), here we take E = 100N/m2_{, ν = 0.3. The computations}

are performed assuming plane strain conditions and either a frictionless (µ = 0) or a frictional (µ = 0.3) contact between the cylinder and the foundation.

In the frictionless case, the analytical solution assuming the linear small strain theory of elasticity is well known. The pressure distribution in the contact region is given by

p(x) = 2F

πb2

p

(a) (b)

(c) (d)

Figure 4: Deformed configuration and contact stress distribution in the patch test. (a)-(b) when the upper body is selected to be contactor, (c)-(d) when the lower body is selected to be contactor (the displacement field is magnified by 100).

R F

x Rigid foundation

Elastic cylinder (E,ν)

(a)

Figure 5: 2D Hertzian contact problem of an elastic cylinder on a rigid foundation. (a) Problem definition, (b) Finer mesh. where b is the half contact region width

b = 2

r

F R(1 − ν2₎

πE (61)

The computation is carried out with two values of the total load: F = 1N/m and F = 2N/m. Due to symmetry, only half of the structure is considered. Two meshes are considered: a coarse one containing 198 elements and 455 nodes, and a finer one containing 345 elements and 773 nodes (the latter is shown in Figure 5b). In both cases, the rigid foundation is chosen as the target and modeled by one single element with fixed nodes. Clearly, the meshes are nonconforming.

The numerical quadrature employs 7 points in the triangles and the elements belonging to the contact line. The penalty parameter is ²N = 100N/m3. The (relative) convergence tolerances are taken to be 10−6for the force residual {R(U, Λ)} =

{Ψ(U)} − {Φ(U)} − {Φcontact(U, Λ)} in (46a) and the multiplier residual {RΛ(U, Λ)} in (46b). Figure 6 plotting the

numerically computed contact pressure acting on the cylinder shows that the finite element solution converges as the mesh is refined.

It should be noted that for a given mesh and a given load, there is in general one element of the cylinder which overlaps the contact area and the free surface, as can be seen in Figure 6. In such an element the contact stress vector - or the multiplier

λ - differs from zero in the contact zone and vanishes outside, and the interpolation of λ from the nodal multipliers, see

Relation (39c), is not precise. Despite this difficulty, the numerical results obtained here are in good agreement with the theory. With the finer mesh and F = 1N/m, the maximal pressure at x = 0 is 2% lower than the theoretical value. With the larger load F = 2N/m, the numerical results deviate more from the theory, the maximal pressure at x = 0 is 2.8% lower than the theoretical value.

In actual fact, the contact stresses obtained here are the nominal Piola-Kirchhoff ones (i.e. the multipliers according to (22)) whereas the analytical pressure in (60) comes from the linear elasticity theory. Although the deformation is small, there

x (m) Ty (N /m 2 ) 0 0.1 0.2 0.3 0.4 0.5 0 0.5 1 1.5 2 2.5 F = 1 N/m F = 2 N/m (a) x (m) Ty (N /m 2 ) 0 0.1 0.2 0.3 0.4 0.5 0 0.5 1 1.5 2 2.5 F = 1 N/m F = 2 N/m (b)

Figure 6: Contact stress distribution in the frictionless Hertzian contact problem. (a) Coarse mesh, (b) Fine mesh.

◦or•: numerical results, solid lines : theoretical results.

still is a slight difference between these stresses. An elaborate calculation shows that the Cauchy stresses corresponding to the numerical Piola-Kirchhoff ones are twice closer to the theoretical values. Having said this, we decide to only present the Piola-Kirchhoff stresses throughout this work.

In the frictional case, the computations are made with the fine mesh only. The analytical solution was obtained by Spence [29, 30], assuming that the influence of friction on the normal stress is negligible in order to uncouple the equations governing the normal and tangential stresses. The contact region is subdivided into a stick zone |x| ≤ c and a slip zone

c ≤ |x| ≤ b. The extent b of the contact region is given by the same relation (61) as in the frictionless case, i.e. b = 0.3404m

if F = 1N/m. The extent c of the stick region is obtained by solving the following relation giving c/b as a function of the coefficient of friction µ and Poisson’s ratio ν

K(c/b) K0_(c/b) = tan−1_µ 1 21+β1−β (62) where β = 1−2ν

2(1−ν), K(k) denotes the complete elliptic integral of the first kind, K(k) =

Rπ/2 0

dφ

√

1−k2_sin2_φ, 0 ≤ k ≤ 1, and

K0_{(k) = K(}√_{1 − k}2_{). With F = 1N/m, µ = 0.3 and ν = 0.3, it is found that c/b = 0.7006, so c = 0.2385m.}

The normal pressure is given by the same relation (60) as in the frictionless case while the tangential contact stresses is computed via q(x) =      2µF x πb2 " 1 K(c/b) Z c/b x/b K(c b, sin−1 tc/b) t2√_{1 − t}2 dt + b c r 1 −³ c b ´2# , |x| ≤ c µ.p(x) , c ≤ |x| ≤ b (63)

where K(k, ψ) is the incomplete elliptic integral of the first kind, K(k, ψ) =R₀ψ √ dφ

1−k2_sin2_φ, 0 ≤ k ≤ 1,

The total load F = 1N/m is applied in 50 equal increments, the penalty parameters are ²N = ²T = 100N/m3. In

each time step, the solution is obtained after 3 iterations in average, with a maximum of 7 iterations at the first time step. The shape of the stresses along the contact interface is displayed in Figure 7a, which clearly shows the progressive transition from stick to slip over the contact line. The normal Tyand tangential Txcontact stresses are plotted in Figure 7b, showing

a good agreement with Spence’s theoretical solution, except in the central stick region where the friction seems to have the effect of raising the normal contact pressure. The value at x = 0 of the normal stress exceeds the analytical one by about 4%, showing the same trend as in [12] where a similar problem of the contact between a rigid punch and an elastic halfspace was investigated.

In order to check whether the finite element procedure enables one to satisfy the local inequality constraints, the ratio

Tx/µTyis computed at every node on the contact line for F = 0.5 and 1N/m and it is verified that the Coulomb inequality

|Tx| ≤ µTyholds everywhere, see Figure 7c. At the sliding nodes, the equality |Tx| = µTyis met within 0.1%.

Eventually, two additional computations are carried out with only one and 10 increments instead of 50. As shown in Figure 8, the contact width, the normal contact pressure as well as the frictional shear in the slip region can be recovered with good precision using one or 10 time steps, whereas the frictional shear distribution in the stick zone is rather sensitive to the load increment and requires up to 50 time steps to be accurate. This result is in full accord with Pietrzak’s observation in [12].

x y 0 0.2 0.4 -10 -9.9 -9.8 -9.7 -9.6 -9.5 (a) F = 1 N/m x (m) C o n ta ct st re ss es (N /m 2 ) 0 0.1 0.2 0.3 0.4 0 0.5 1 1.5 2 slip stick

T

T

T

=

µ

.T

F = 1 N/m

F = 1 N/m

F = 0.5 N/m

Figure 7: Contact stress distribution in the frictional Hertzian contact problem (F = 1N/m, µ = 0.3).

(a) Spatial distribution along the contact line, (b)◦or•: numerical results, solid lines : theoretical results. (c) Check of the Coulomb friction law (F = 0.5 and 1N/m).

x (m) C o n ta ct st re ss es (N /m 2) 0 0.1 0.2 0.3 0.4 0 0.5 1 1.5 2 10 time steps T_y: 1 time step T_x: 1 time step 10 time steps

6.3 Contact between two beams

In this example, we are concerned with the quasistatic contact between two beams, one of which being subjected to either a dead or a follower force, with a view to showing the ability of the proposed formulation to deal with large deformation contact. The two beams have identical reference geometries, with rectangular cross section, a length of L = 1m along x, a height of 0.05m along y and a thickness of 0.1m along z, see Figure 9. They are both made of an isotropic hyperelastic Saint-Venant Kirchhoff material with E = 2.106_N/m2_{and ν = 0.3.}

The beams are clamped at end X = 0. In the reference configuration, the lower face of the upper beam (contactor) coincide with the upper face of the lower beam (target).

X Y Z (a) Upper beam Lower beam 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -0.2 -0.1 0 0.1 y X Y Z (b) A

Figure 9: Reference configuration of the two beam problem. (a) 3D view of the mesh (b) x-y in-plane view. Each beam is modeled with 15 20-node brick (i.e. hexahedral) elements (H20) and the contact surface with 15 8-node quadrilaterals (Q8). The numerical quadrature employs 3 × 3 × 3 Gauss points in the H20’s and 6 × 6 points in the Q8 elements belonging to the contact surface.

In the following, two types of loading, a dead and a follower force, are applied on the contactor. For each loading, the computations are carried out assuming either frictionless or frictional contact, µ = 0.3. In any case, the penalty parameters are ²N = ²T = 105N/m3.

6.3.1 Dead load

A dead surface load −F y is uniformly distributed over the cross section X = 1m of the upper beam. Force F is progressively increased from zero using an arc length method with the arc length regularly incremented by a factor 1.1. The (relative) convergence tolerances are taken to be 10−6_{for the force residual {R(U, Λ)} in (46a), 10}−6_{for the arc-length residual and}

10−10_{for the multiplier residual {R}

Λ(U, Λ)} in (46b).

In the frictionless case, the computation is carried out in 29 steps, the resultant force at the last step is F = 62.62N and the convergence is achieved in 7-8 iterations in average. Figure 10 shows the deformed configurations and the contact traction vectors acting on the upper beam. Only x-y in-plane views are shown since the complete 3D views are not legible. In the frictional case, the entire calculation requires 38 steps, the final resultant force is F = 61.20N and the convergence achieved in 11-12 iterations in average. The corresponding deformed configurations and the contact tractions are shown in Figure 11.

The contact immediately takes place after force F is applied and persists over the entire lower surface of the upper beam until the last step. During the first steps, the contact tractions are higher near the end cross section where force F is applied, whereas far from this section, the contact tractions are smaller and barely visible in Figures 10 and 11. As the deformations grow up, the contact tractions become significant in the neighborhood of the built-in section. It should be mentioned that there are possible spatial oscillations of the contact stresses due to the use of quadratic shape functions which take negative values at certain points and due to coarse meshes. Thus, during the bending process, there may be some nodes on the contact surface where the contact law is not satisfied, namely TN < 0 or kTTk > µ.TN in frictional case. Fortunately, these

violations only occur at the nodes where TN and kTTk are very small, specifically smaller than about 1% of the maximal

TN and kTTk values on the whole contact surface. Moreover, the negative contact pressures disappear if the mesh is fine

enough. In Figures 10 and 11, the contact tractions violating the contact law are hardly visible. In presence of the friction, the contact state is rather complex. Roughly speaking, it is found that there is slip at most nodes and every step, and stick at some nodes and some steps.

Figure 12 shows that the deflection at point A (central point of the cross section X = 1m of the upper beam, as shown in Figure 9) versus force F is almost the same, with or without friction. If the friction prevented the slip to occur at some

X Y Z (a) F X Y Z (b) X Y Z (c) X Y Z (d) X Y Z (e) X Y Z (f)

Figure 10: Contact between two beams. Dead load and frictionless case. (a) F = 1.625N , (b) F = 6.914N , (c) F = 17.17N , (d) F = 29.21N , (e) F = 41.11N , (f) F = 62.62N . The dead load F is not drawn to scale.

X Y Z F (a) X Y Z (b) X Y Z (c) X Y Z (d) X Y Z (e) X Y Z (f)

Figure 11: Contact between two beams. Dead load and frictional case, µ = 0.3. (a) F = 1.769N , (b) F = 6.914N , (c)

F = 19.78N , (d) F = 34.36N , (e) F = 49.28N , (f) F = 61.20N . The dead load F is not drawn to scale.

steps, force F necessary to reach the same deflection would be higher than in the frictionless case and the two plots would be distinct.

v/L F L 2 /E I -1 -0.75 -0.5 -0.25 0 0 10 20 30

Figure 12: Contact between two beams. Deflection at point A versus dead load F (I = the second moment of area of the upper beam). ◦: no friction,•: with friction.

6.3.2 Follower force

The dead force considered above is now replaced with a uniform pressure distributed over the last Q8 element of the upper face of the upper beam, next to the cross section X = 1m. The pressure is a follower force in the sense that it remains continually perpendicular to the area on which it is applied, so that its direction changes as the beam deforms, contrary to the dead force which keeps a fixed direction. We choose the same automatic control of the pressure and convergence tolerances as in the dead load case.

In the frictionless case, the computation is carried out in 34 steps leading to the final pressure resultant P = 73.84N , the convergence is achieved in 9-10 iterations in average and the deformed configurations together with the contact stresses are shown in Figure 13. In the frictional case (µ = 0.3), the computation is carried out in 43 steps, the final pressure resultant is

P = 74.83N , the convergence is achieved in 10-11 iterations in average and the results are shown in Figure 14.

As shown in Figure 13, at the beginning the contact occurs on the whole lower surface of the upper beam. At an advanced deformed state, the absence of friction leads to a release over a large portion of the former contact surface, as shown in Figures 11e and 11f. On the other hand, in the presence of sufficiently high friction, the contact is maintained all over the lower surface of the upper beam, as shown in Figure 14.

As in the dead load case, the contact state in the presence of friction is rather complex. It is found that at every step there is slip at most nodes, except the last step where there is more stick than slip.

Figure 15 shows the deflection at point A of the upper beam versus pressure resultant P . Note that under the follower loading there is a limit point for the deflection, the maximum absolute value for the deflection being equal to about three-quarter of the beam length.

X Y Z P (a) X Y Z (b) X Y Z (c) X Y Z (d) X Y Z (e) X Y Z (f)

Figure 13: Contact between two beams. Follower load and frictionless case. (a) P = 1.607N , (b) P = 6.757N , (c)

P = 18.88N , (d) P = 33.35N , (e) P = 46.44N , (f) P = 73.84N . The follower load is not drawn to scale.

X Y Z P (a) X Y Z (b) X Y Z (c) X Y Z (d) X Y Z (e) X Y Z (f)

Figure 14: Contact between two beams. Follower load and frictional case, µ = 0.3. (a) P = 1.757N , (b) P = 6.743N , (c)

v/L P L 2 /E I -1 -0.75 -0.5 -0.25 0 0 10 20 30 40

Figure 15: Contact between two beams. Follower load. Deflection at point A versus pressure resultant P . ◦: no friction,•: with friction.